Soft characterisations of regular ordered semigroups

Let U be an initial universe set and let S be an ordered semigroup. The concepts of int-soft left, int-soft right ideals, int-soft quasi-ideal and int soft bi-ideal on S were introduced in [5]. In this paper, we study these ideas by providing an equivalent definition of soft right, soft left ideals and soft quasi-ideals. Based on this idea, we show that S is regular if and only if ( α ◦ β, S ) = ( α (cid:101) β, S ) for every soft right ideal ( α, S ) and every soft left ideal ( β, S ) over U , and the soft right and the soft left ideals over U are idempotent and for each soft right ideal ( α, S ) and each soft left ideal ( β, S ) over U , the soft set ( α ◦ β, S ) is a soft quasi-ideal over U .